Let us summarize the theory of [1]}, rephrasing it in line with our approach to \(\operatorname{Eig}(V)\) . First, the state space \(V\)  is now assumed to be a Hilbert space, i.e. \(V\)  should be equipped with a Hermitian inner product \(\langle \;|\;\rangle \) . This allows one to define the unit sphere \(S_1(V)\)  inside \(V\)  by restricting to norm 1 states. As norm 1 fixes a state up to a \(U(1)\) -phase, \(S_1(V)\)  is naturally a \(U(1)\) -manifold, and the quotient is the projective space \(P(V)\)  of \(V\) . This quotient defines the principal bundle
\( \begin{tikzcd}U(1) {r} & S_1(V) {r} & P(V)\end{tikzcd}\) 
